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The modified Helmholtz equation appears in many areas of engineering and science[1],such as in implicit marching schemes for the heat equation, in Debye-Huckel theory, in the linearization of the Poisson-Boltzmann equation and so on [5, 8, 16]. The direct problems for the modified Helmholtz equation have been studied extensively in the past century [19, 7]. However, in some practical problems, the boundary data on the whole boundary cannot be obtained. We only know the noisy data on a part of the boundary or at some interior points, giving rise to an inverse problem [6]. The Cauchy problem for the modified Helmholtz equation is an inverse problem and is typically ill-posed. It means that a small change in the Cauchy data may result a dramatic change in the solution. Therefore, it is necessary to study different highly efficient algorithms for solving it. In recent years, there are many special numerical methods to deal with this problem, such as the boundary element method[9], the method of fundamental solutions[10, 17], the conjugate gradient method[11], the Landweber method[9], quasi-reversibility and truncation method[14], quasi-boundary and Tikhonov type regularization method[13, 18], the Fourier regularization method [3, 4] and so on[15]. In the present paper we will consider the following problem with inhomogeneous Dirichlet data in a strip domain:

Optimal Error Bound and A Semi-discrete Difference Scheme for the Cauchy Problem of the Modified Helmholtz Equation